Question

How do you determine at which the graph of the function `y=1/x^2` has a horizontal tangent line?

Answer 1

By using derivatives

Explanation:

derivatives define the slope of a tangent line at a point on the function
therefore if the tangent line is horizontal, its slope is 0

so, on differentiating
`y'(x) = d/dx 1/x^2= 0`

we're setting it equal to zero because we want to see the points at which the derivative is 0 so its slope Is 0

we can use the power rule here as `1/x^2` is just `x^-2`

therefore,

`-2x^-3 = 0`
divide both sides by `-2`

`= 1/x^3 = 0`

this is an indeterminate form as the only way this could satisfy the equation is if x was positive or negative infinity
therefore at finite values of x, we don't ever have a point where the tangent lines are horizontal

you can see this on the graph that as x becomes bigger and bigger its slope decreases and gets closer and closer to 0, so as x approaches infinity, its slope approaches 0

graph{1/x^2 [-12.66, 12.65, -6.33, 6.33]}